Question

The direction ratio of the normal to the plane passing through the points $(1,2,-3),$ $(-1,-2,1)$ and parallel to $\frac{x-2}{2}=\frac{y+1}{3}=\frac{z}{4}$ is

• A. $(2,3,4)$
• B. $(14,-8,-1)$
• C. $(-2,0,-3)$
• D. $(1,-2,-3)$

Answer 1

The correct option is B $(14,-8,-1)$

Direction ratio of normal vector of plane is given by cross product of direction ratios of two non parallel vector in the plane.
$A=(1,2,-3)$
$B=(-1,-2,1)$

One vector is $\overrightarrow{BA}=2\hat{i}+4\hat{j}-4\hat{k}$
Another is given by the dr of the line parallel to the plane.
$\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& 4\\ 1+1& 2+2& -3-1\end{array}\right|=-28\hat{i}+16\hat{j}+2\hat{k}=-2(14\hat{i}-8\hat{j}-1\hat{k})$

Direction ratios of the normal to the plane can be $(14,-8,-1)$